ASSESMENT OF SMARANDACHE CURVES IN THE NULL CONE Q2

arXiv:1703.07664v1 [math.GM] 7 Mar 2017

ASSESMENT OF SMARANDACHE CURVES IN THE NULL CONE Q2 MIHRIBAN KULAHCI AND FATMA ALMAZ Abstract. In this paper, we give Smarandache curves according to the asymptotic orthonormal frame in null cone Q2 . By using cone frame formulas, we present some characterizations of Smarandache curves and calculate cone frenet invariants of these curves. Also, we illustrate these curves with an example.

1. Introduction Human being were bewitched by curves and curved shapes long before they took into account them as mathematical objects. But the greatest effect in the research of curves was, of course, the discovery of the calculus. Geometry before calculus includes only the simplest curves.

In classical curve theory, the geometry of a curve in three-dimensions is actually characterized by Frenet vectors.

Smarandache geometry is a geometry which has at least one Smarandachely denied axiom [4]. An axiom is said to be Smarandachely denied, if it behaves in at least two different ways within the same space. Smarandache curve is defined as a regular curve whose position vector is composed by Frenet frame vectors of another regular curve. Smarandache curves in various ambient spaces have been classfied in [1]-[14], [19], [21]-[32].

In this study, we define special Smarandache curves such as xα, xy, αy and xαy Smarandache curves according to asymptotic orthonormal frame in the null cone Q2 and we examine the curvature and the asymptotic orthonormal frame’s vectors of Smarandache curves. We also give an example related to these curves.

2. Preliminaries Some basics of the curves in the null cone are provided from, [15]-[18].

Let E13 be the 3−dimensional pseudo-Euclidean space with the g(X, Y ) = hX, Y i = x1 y1 + x2 y2 − x3 y3 e

for all X = (x1 , x2 , x3 ), Y = (y1 , y2 , y3 ) ∈ E13 . E13 is a flat pseudo-Riemannian manifold of signature (2, 1).

Date: 2017.

2000 Mathematics Subject Classification. Primary 53A40; Secondary 53A35.

Key words and phrases. Smarandache curve, asymptotic orthonormal frame, null cone, cone frame formulas.

This paper is in final form and no version of it will be submitted for publication elsewhere.

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MIHRIBAN KULAHCI AND FATMA ALMAZ

Let M be a submanifold of E13 . If the pseudo-Riemannian metric ge of E13 induces a pseudo-Riemannian metric g(respectively, a Riemannian metric, a degenerate quadratic form) on M , then M is called a timelike( respectively, spacelike, degenerate) submanifold of E13 . Let c be a fixed point in E13 . The pseudo-Riemannian lightlike cone(quadric cone ) is defined by

Q21 (c) = x ∈ E13 : g(x − c, x − c) = 0 ,

where the point c is called the center of Q21 (c). When c = 0, we simply denote Q21 (0) by Q2 be and call it the null cone.

Let E13 be 3-dimensional Minkowski space and Q2 the lightlike cone in E13 . A vector V 6= 0 in E13 is called spacelike, timelike or lightlike, if hV, V i > 0, hV,pV i < 0 or hV, V i = 0, respectively. The norm of a vector x ∈ E13 is given by kxk = hx, xi, [20].

We assume that curve x : I → Q2 ⊂ E13 is a regular curve in Q2 for t ∈ I. In the following, we always assume that the curve is regular.

A frame field {x, α, y} on E13 is called an asymptotic orthonormal frame field, if hx, xi = hy, yi = hx, αi = hy, αi = 0, hx, yi = hα, αi = 1.

Using x′ (s) = α(s) we know that {x(s), α(s), y(s)} from an asymptotic orthonormal frame along the curve x(s) and the cone frenet formulas of x(s) are given by x′ (s) = α(s) (2.1)

α′ (s) = κ(s)x(s) − y(s) y ′ (s) = −κ(s)α(s)

where the function κ(s) is called cone curvature function of the curve x(s), [17].

Let x : I → Q2 ⊂ E13 be a spacelike curve in Q2 with an arc length parameter s. Then x = x(s) = (x1 , x2 , x3 ) can be written as fs−1 2 (f − 1, 2f, f 2 + 1) 2 for some non constant function f (s) and fs = f ′ , [18].

(2.2)

x(s) =

3. Smarandache Curves in The Null Cone Q2 In this section, we define the Smarandache curves according to the asymptotic orthonormal frame in Q2 . Also, we obtain the asymptotic orthonormal frame and cone curvature function of the Smarandache partners lying on Q2 using cone frenet formulas.

Smarandache curve γ = γ(s∗ (s)) of the curve x is a regular unit speed curve lying fully on Q2 . Let {x, α, y} and {γ, αγ , yγ } be the moving asymptotic orthonormal frames of x and γ, respectively.

Definition 1. Let x be unit speed spacelike curve lying on Q2 with the moving asymptotic orthonormal frame {x, α, y} . Then, xα−smarandache curve of x is defined by c (3.1) γxα (s∗ ) = x(s) + α(s), b

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where c, b ∈ R+ 0.

Theorem 1. Let x be unit speed spacelike curve in Q2 with the moving asymptotic orthonormal frame {x, α, y} and cone curvature κ(s) and let γxα be xα−smarandache curve with asymptotic orthonormal frame {γxα , αxα , yxα } . Then the following relations hold: i) The asymptotic orthonormal frame {γxα , αxα , yxα } of the xα-smarandache curve γxα is given as     c 1 0 x γxα b αxα  =  ψ b κψ − b ψ  α (3.2) c c y yxα ̺1 ̺2 ̺3 where (3.3)

(3.4)

and

c , ψ= √ c2 − 2κb2 b b b2 2 Υ1 = ( κ′ + κ)ψψ ′ + 2 κ (ψ ′ ) , c c c b b Υ2 = ψ(ψ ′ + 2 κψ), c c b b Υ3 = − ψ( ψ ′ + ψ) c c

c c 2Υ1 Υ3 + Υ22 ) = −Υ1 + κγxα (s∗ ), 2b b

1 ̺2 = −(Υ2 + 2Υ1 Υ3 + Υ22 ) = −Υ2 + κγxα (s∗ ), 2 ̺3 = −Υ3 . ̺1 = −(Υ1 +

(3.5)

ii) The cone curvature κγxα (s∗ ) of the curve γxα is given by

1 (3.6) κγxα (s∗ ) = − 2Υ1 Υ3 + Υ22 .

2 where Z 1 p 2 s∗ = c − 2b2 κ(s)ds. b

Proof. i) We assume that the curve x is a unit speed spacelike curve with the asymptotic orthonormal frame {x, α, y} and cone curvature κ. Differentiating the equation (3.1) with respect to s and considering (2.1), we have

b b ′ (3.7) γxα (s∗ ) = A α(s) + κx(s) − y(s) , c c where 1p 2 ds∗ = (3.8) c − 2b2 κ(s), ds b c (3.9) . ψ=p 2 c − 2κ(s)b2

′ It can be easily seen that the tangent vector γxα (s∗ ) = αxα (s∗ ) is a unit spacelike vector.

Differentiating (3.7) , we obtain equation as follows

(3.10)

′′ γxα (s∗ ) = Υ1 x(s) + Υ2 α(s) + Υ3 y(s),

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MIHRIBAN KULAHCI AND FATMA ALMAZ

where b2 b b 2 Υ1 = ( κ′ + κ)ψψ ′ + 2 κ (ψ ′ ) , c c c b b Υ2 = ψ(ψ ′ + 2 κψ), c c b b ′ Υ3 = − ψ( ψ + ψ). c c (3.11)

′′ yxα (s∗ ) = −γxα −

1 ′′ hγ , γ ′′ i γxα 2 xα xα

By the help of previous equation (3.11), we obtain yxα (s∗ ) = ̺1 x(s) + ̺2 α(s) + ̺3 y(s),

c 2Υ1 Υ3 + Υ22 ), ̺2 = −(Υ2 + 21 2Υ1 Υ3 + Υ22 ), ̺3 = −Υ3 . where ̺1 = −(Υ1 + 2b ii) The curvature κγxα (s∗ ) of the γxα (s∗ ) is explicity obtained by (3.12)

1 ′′ ′′ κγxα (s∗ ) = − hγxα , γxα i 2

1 (3.13) = − 2Υ1 Υ3 + Υ22 .

2 Thus, the theorem is proved.

Definition 2. Let x be unit speed spacelike curve lying on Q2 with the moving asymptotic orthonormal frame {x, α, y} . Then, xy−smarandache curve of x is defined by (3.14)

1 γxy (s∗ ) = √ (cx(s) + by(s)) , 2cb

where c, b ∈ R+ 0.

Theorem 2. Let x be unit speed spacelike curve in Q2 with the moving asymptotic orthonormal frame {x, α, y} and cone curvature κ and let γxy be xy−smarandache curve with asymptotic orthonormal frame {γxy , αxy , yxy } . Then the following relations hold: i) The asymptotic orthonormal frame {γxy , αxy , yxy } of the xy−smarandache curve γxy is given as   p q     c b 0 γxy 2b 2c  x    αxy  =  0 (3.15) 1 0   α .  2√ √ bκ 2bc c 2bc y yxy 0 (c−bκ)2 (c−bκ)2 ii) The cone curvature κγxy (s∗ ) of the curve γxy is given by

(3.16)

κγxy (s∗ ) =

where (3.17)

∗

s =√

1 2cb

Z

2bcκ(s)

2,

(c − bκ(s))

(c − bκ(s)) ds.

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Proof. i) We assume that the curve x is a unit speed spacelike curve with the asymptotic orthonormal frame {x, α, y} and cone curvature κ. Differentiating the equation (3.14) with respect to s and considering (2.1), we have (3.18)

′ γxy (s∗ )

1 ds∗ → =√ (c − bκ(s)) − α (s). ds 2cb

By considering (3.17), we get (3.19)

′ γxy (s∗ ) = α(s) = αxy .

→ Here, it can be easily seen that the tangent vector − α xy is a unit spacelike vector.

(3.20)

′′ γxy (s∗ )

ds∗ = κx(s) − y(s). ds

By substituting (3.17) into (3.20) and making necessary calculations, we obtain √ √ κ 2bc − 2bc − → → ′′ ∗ x − (3.21) γxy (s ) = 2 y . (c − bκ) (c − bκ)

′′ ′′ ′′ By the help of equation yxy (s∗ ) = −γxy − 21 γxy , γxy γxy , we write √ √ c 2bc b 2bc.κ2 x(s) + y(s).

(3.22) yxy (s∗ ) = (c − bκ)2 (c − bκ)2 ii) The curvature κγxy (s∗ ) of the γxy (s∗ ) is explicity obtained by κγxy (s∗ ) = −

2bcκ(s) 1 ′′ ′′ . = γ ,γ 2 xy xy (c − bκ(s))2

Definition 3. Let x be unit speed spacelike curve lying on Q2 with the moving asymptotic orthonormal frame {x, α, y} . Then, αy−smarandache curve of x is defined by (3.23)

b γαy (s∗ ) = α(s) + y(s), c

where c, b ∈ R+ 0.

Theorem 3. Let x be unit speed spacelike curve in Q2 with the moving asymptotic orthonormal frame {x, α, y} and cone curvature κ and let γαy be αy−smarandache curve with asymptotic orthonormal frame {γαy , ααy , yαy } . Then the following relations hold: i) The asymptotic orthonormal frame {γαy , ααy , yαy } of the αy−smarandache curve γαy is given as      b 0 1 γαy x c √ √ √   c κ b κ c κ ααy  =  √ √ √ α (3.24) ,  b2 −2c2  b2 −2c2 κ b2 −2c2 yαy y ω1 ω2 ω3

6

where

(3.25)

and

MIHRIBAN KULAHCI AND FATMA ALMAZ

cκ′ c − 2κb ζ1 = 2 , b − 2c2 2κ

c−b−1 cκ′ , ζ2 = 2 2 b − 2c 2κ

cκ′ bκ − c ζ3 = 2 2 b − 2c 2κ2 ω1 = −ζ1 ,

1 2ζ1 ζ3 + ζ22 ), 2

b 2ζ1 ζ3 + ζ22 ). ω3 = −(ζ3 + 2c ii) The cone curvature κγαy (s∗ ) of the curve γαy is given by

κ′ 2 1 c2 2 ∗ ( ) (3.27) κγαy (s ) = − (c − 2κb) (bκ − c) + (c − b − 1) , κ 8 (b2 − 2c2 )2 κ where √ Z p b2 − 2c2 ∗ κ(s)ds.

(3.28) s = c (3.26)

ω2 = −(ζ2 +

Proof. i) Let the curve x be a unit speed spacelike curve with the asymptotic orthonormal frame {x, α, y} and cone curvature κ. Differentiating the equation (3.23) with respect to s and considering (2.1), we find −−→ b −−→ −−→ ds∗ ′ = κx(s) − κα(s) − y(s). γαy (s∗ ) ds c This can be written as following −−→ b −−→ −−→ ds∗ (3.29) ααy (s∗ ) = κx(s) − κα(s) − y(s), ds c where √ ds∗ b2 − 2c2 p (3.30) = κ(s). ds c By substituting (3.30) into (3.29), we find √ √ √ b κ − c κ c κ − → → → − ∗ x −√ α − √ y.

(3.31) ααy (s ) = √ b2 − 2c2 b2 − 2c2 κ b2 − 2c2 Differentiating (3.31) and using (3.30), we get

′′ γαy (s∗ ) = ζ1 x(s) + ζ2 α(s) + ζ3 y(s),

′ ′ ′ c−2κb c−b−1 where ζ1 = b2cκ , ζ2 = b2cκ , ζ3 = b2cκ −2c2 2κ −2c2 2κ −2c2

1 ′′ ′′ γ , γ ′′ γαy .

(3.32) yαy (s∗ ) = −γαy − 2 αy αy By the help of equation (3.32), we obtain

bκ−c 2κ2

.

yαy (s∗ ) = ω1 x(s) + ω2 α(s) + ω3 y(s),

b where ω1 = −ζ1 , ω2 = −(ζ2 + 12 2ζ1 ζ3 + ζ22 ), ω3 = −(ζ3 + 2c 2ζ1 ζ3 + ζ22 ).

(3.33)

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ii) The curvature κγαy (s∗ ) of the γαy (s∗ ) is explicity obtained by ′ 2

κ c2 (c − 2κb) (bκ − c) 2 ∗ κγαy (s ) = − + (c − b − 1) .

2 κ κ 8 (b2 − 2c2 )

7

Definition 4. Let x be unit speed spacelike curve lying on Q2 with the moving asymptotic orthonormal frame {x, α, y} . Then, xαy−smarandache curve of x is defined by 1 γxαy (s∗ ) = √ (cx(s) + bα(s) + c∗ y(s)) , 2cc∗ + b2

(3.34)

where c, c∗ , b ∈ R+ 0.

Theorem 4. Let x be unit speed spacelike curve in Q2 with the moving asymptotic orthonormal frame {x, α, y} and cone curvature κ and let γxαy be xαy−smarandache curve with asymptotic orthonormal frame {γxαy , αxαy , yxαy } . Then the following relations hold: i) The asymptotic orthonormal frame {γxαy , αxαy , yxαy } of the xαy−smarandache curve γxαy is given as      ∗ √ c √ b √ c x γxαy 2cc∗ +b2 2cc∗ +b2 2cc∗ +b2    αxαy  =  ρ1 α (3.35) ρ2 ρ3 y yxαy σ1 σ2 σ3

where

η=

q (c − c∗ κ(s))2 − 2b2 κ(s)

bκ(s) c − c∗ κ(s) b , ρ2 = , ρ3 = − η η η ξ1 = (ρ′1 + ρ2 κ) , ξ2 = ρ′2 + ρ1 + ρ3 κ, ξ3 = −ρ′3 − ρ2

ρ1 =

(3.36)

and

c σ1 = −ξ1 − √ 2ξ1 ξ3 + ξ22 , ∗ 2 2 2cc + b

b 2ξ1 ξ3 + ξ22 , σ2 = −ξ2 − √ ∗ 2 2 2cc + b

c∗ 2ξ1 ξ3 + ξ22 . σ3 = −ξ3 − √ ∗ 2 2 2cc + b

(3.37)

ii) The cone curvature κγxαy (s∗ ) of the curve γxαy is given by ! ! ′ ′ ∗ ∗ c − c κ c − c κ κ b κγyxα (s∗ ) = b + + κ η η η η

′ 1 c − c∗ κ (3.38) . − 2 η where (3.39)

1 s = √ 2cc∗ + b2 ∗

Z q 2 (c − c∗ κ(s)) − 2b2 κ(s)ds, b, c, c∗ ∈ R+ 0.

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MIHRIBAN KULAHCI AND FATMA ALMAZ

Proof. i) Differentiating the equation (3.34) with respect to s and considering (2.1), we find −−→ −−→ −−−→ ds∗ 1 ′ (3.40) γxαy (s∗ ) bκx(s) + (c − c∗ κ) α(s) − by(s) . =√ ds 2cc∗ + b2

This can be written as follows (3.41)

αxαy (s∗ ) =

bκ −−→ c − c∗ κ −−→ b −−→ x(s) + α(s) − y(s), η η η

or (3.42)

−−→ −−→ −−→ αxαy (s∗ ) = ρ1 x(s) + ρ2 α(s) − ρ3 y(s).

where (3.43)

ds∗ 1 = √ ds 2cc∗ + b2

q 2 (c − c∗ κ) − 2b2 κ

Differentiating (3.42) and using (3.43), we get ′′ γxyα (s∗ ) = ξ1 x(s) + ξ2 α(s) + ξ3 y(s),

where ξ1 = (ρ′1 + ρ2 κ) , ξ2 = ρ′2 + ρ1 + ρ3 κ, ξ3 = −ρ′3 − ρ2 .

(3.44)

′′ yxαy (s∗ ) = −γxαy −

By the help of equation (3.44), we obtain

1 ′′ γxαy . γ , γ ′′ 2 xαy xαy

yxay (s∗ ) = σ1 x(s) + σ2 α(s) + σ3 y(s),

where σ1 = −ξ1 − 2√2ccc∗ +b2 2ξ1 ξ3 + ξ22 , σ2 = −ξ2 − 2√2ccb∗ +b2 2ξ1 ξ3 + ξ22 , σ3 =

c∗ 2ξ1 ξ3 + ξ22 . −ξ3 − 2√2cc ∗ +b2

′′ ′′ ii) From κγxαy (s∗ ) = − 12 γxαy , we have , γxαy ! ! ′

′ ′ κ 1 c − c∗ κ c − c∗ κ c − c∗ κ b ∗ κγyxα (s ) = b − + + . κ η η η η 2 η (3.45)

Theorem 5. Let x : I → Q2 ⊂ E13 be a spacelike curve in Q2 as follows (3.46)

x(s) =

fs−1 2 (f − 1, 2f, f 2 + 1), 2

for some non constant function f (s). Then we can write the following statements: • If x is a xα− smarandache curve, then the xα− smarandache curve γxα can be written as c (3.47) γxα (s∗ ) = ( − fs−1 fss )x(s) + (f, 1, f ) . b • If x is a xy− smarandache curve, then the xy− smarandache curve γxy can be written as

1 1 −2 2 ∗ −1 (3.48) γxy (s ) = √ (c − fs fss )x(s) + fs fss (f, 1, f ) − fs (1.0, 1) .

2 2bc

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• If x is a αy− smarandache curve, then the αy− smarandache curve γαy can be written as

b −1 b −2 2 ∗ −1 (3.49) γαy (s ) = −fs fss − fs fss x(s) + 1 + fs fss (f, 1, f ) 2c c b − fs (1.0, 1) . c • If x is a xαy− smarandache curve, then the xαy− smarandache curve γxαy can be written as

!

∗ 2 1 x(s) c − bfs−1 fss − c2 fs−2 fss ∗ , γxαy (s ) = √

2cc∗ + b2 + b + c∗ fs−1 fss (f, 1, f ) − fs (1.0, 1) where c, b, c∗ ∈ R+ 0.

Proof. It is obvious from (3.1), (3.14), (3.23), (3.34) and (3.46).

We can give the following example to hold special Smarandache curves in the null cone Q2 . xα, xy, αy, and xαy special smarandache curves of x curves are given in Figure 1 A, C, E, G, I, respectively. These figures rotated in three dimensions are also given in Figure 1 B, D, F, H, J, respectively.

Example 1. The curve x(s) =

1 cosh s cosh s − , tanh s, 2 cosh s 2

is spacelike in Q2 with arc length parameter s. Also, the shape of the x curve is given as follows Then we can write the smarandache curves of the x-curve as follows: i) xα− smarandache curve γxα is given by sinh s tanh s   s 1 d cosh − cosh − cosh s , 2 s + 2 1  d tanh s + cosh γxα (s) =  2 s, cosh s sinh s d 2 + 2

ii) xy− smarandache curve γxy is given by 

2 s m cosh s − n tanh s sinh s − d 1c+tanh ,  cosh s

 tanh2 s γxy (s) =  , c tanh s e − 2 

2 1 s cosh s − d c − tanh 2 2 cosh s

    

iii) αy− smarandache curve γαy is given by

 cosh s 1 − cosh 2 s (1 − e tanh s) tanh s + sinh s (1 + d tanh s)  −d cosh s, γαy (s) =   − tanh2 s (1 + d tanh s) + d tanh s, sinh s − d cosh s − e sinh s tanh s 2

   

iv) xαy− smarandache curve γxαy is given by   b sinh s−c∗ tanh2 s c , m sinh s − n cosh s + d sinh s tanh s − cosh s − cosh2 s  γxαy (s) =  m + n tanh s + k tanh2 s + l tanh3 s, m cosh s + d sinh s + c tanh s sinh s, where m, n, d, c, k, l, e ∈ R+ 0.

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MIHRIBAN KULAHCI AND FATMA ALMAZ

(a) The curve x

(b) The rotated surface of curve x

(c) xα−smarandache curve

(d) xα- smarandache surface.

(e) xy−smarandache curve

(f) xy−smarandache surface.

(g) αy−smarandache curve.

(h) αy−smarandache surface.

(i) xαy−smarandache curve.

(j) xαy−smarandache surface.

Figure 1

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MIHRIBAN KULAHCI AND FATMA ALMAZ

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Department of Mathematics, Firat University, 23119 ELAZIĞ/TÜRKİYE E-mail address: mihribankulahci@gmail.com E-mail address: fb fat almaz@hotmail.com